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"(The simple diagram only works if we choose units for sheep, meat, and wool so that 1 unit of sheep produces 1 unit of meat and 1 unit of wool, otherwise we need a little bit of simple algebra to do the same analysis.)"

I think your diagram only works if 1 unit of sheep produces 1 unit of meat OR 1 unit of wool. If it's "AND" then the demand for sheep will equal the larger of the two individual demands for meat and wool, not the sum.

Why only one output? The answer is IMO simple: ease of visualization.

two hats: I think it's the other way around. If one sheep produces one unit of meat OR one unit of wool, we add the two demand curves up horizontally to get the total demand for sheep. And the price of meat would be equal to the price of wool (as long as some of each is being bought in equilibrium).

And all (I think) textbooks show how and when to add up demand curves horizontally. We add up the demand curves of all the individual consumers of meat horizontally to get the market demand curve for meat.

anon: OK. Maybe. But wouldn't that also predict we would talk about only one input, to make it easier to visualise? Maybe we do: we use the short run/long run trick, to disappear one of the inputs.

"why do we seem to spend so much time talking about multiple inputs and one output and so little time talking about one input and multiple outputs?"

Is it because it appears to us that we have to make fewer assumptions about preferences for the former than the latter? Take a given set of costs and a particular output; variations in costs due to pleasingly cross-cultural factors like input costs or technology or wages or energy prices move the slope of the supply curve to the left or right, altering the price. So we talk usefully model any human culture, from the Sentinelese to movie stars, and our epsitemic limitations are a similarly cross-cultural set of known unknowns that are the same for any particular culture e.g. the cost of grain for a miller in his supply-curve for bread.

With the latter, you have the same input having multiple outputs that clearly depend on culture and/or entrepreneurship. Sticking with sheep: some cultures eat sheep's eyes and brains, some don't. A vegetarian commune might keep sheep for wool, but not for meat. An entrepreneur might see a product with a standard use and then find an alternative use where it can be sold for a higher price e.g. Listerine as mouthwash rather than disinfectant. The kinds of assumptions one makes when drawing a graph like that output graph for sheep become embarassing. Talking about single input/multiple outputs without bringing culture, entrepeneurship, advertising etc. into the picture just seems bizarre.

So is it that the idealisations involved in a multiple inputs/single output model seem less severe? Or perhaps just more familiar?

W Peden: Interesting theory. Not obviously wrong. If true, I might re-describe it as: because we are biased against subjectivist theory of value??

Question for historians of thought: which economist(s) first talked about the meat/wool example (or similar)? And when?

Pure guess: Jevons.

Nick Rowe,

I think we're all biased against the subjectivist theory of value to some extent, but I'm not sure if that's the cause of the disinclination. Or, put another way, maybe economists are cautious in the face of dispersed knowledge (which is what the importance of culture/entrepeneurship/advertising et al comes down to here) but at the end of the day you still have to talk about supply and demand curves, so the paradigmatic cases will be those that require the least severe idealisations.

I have trouble formulating it, but I think there's a connection with complementary goods and substitutable goods here. Why did I learn about substitutable goods the first time I snuck into an intro to economics lecture, but it was only in the last few hours that I came across the term "complementary goods"? Idealising the relationship between loaves of bread and containers of butter seems much more questionable than idealising the relationship between PCs and Macs.

My standard graduate course in micro taught pretty much all production theory in terms of "netput vectors" i.e. a vector of outputs minus a vector of inputs. We used Varian (which, in my ignorance, I think has been the most widely used graduate micro textbook for a couple of decades.)

I thought only macroeconomists thought in terms of a single produced good.

W. Peden: Hmmm. There are substitutes and complements in demand, and substitutes and complements in supply. Just a random thought. I'm still mulling over your comment.

Speaking of the history of economic thought, IIRC atleast two early schools of thought preferred the paradigm of one input - many outputs. I think Quesnay and the Physiocrats typically thought of land as the single input (which produced everything else) while Marx used labour as the single input.

IIRC Varian's grad text does deal with the production set, but first deals with firms that produce only one output. Maybe this is because a transformation function is more difficult for most students to grasp than a production function?

Yes, of course. I had flipped the axes in my head when I made the comment (ie, quantity on the y-axis). And actually, something different from price on the x (what, I'm not sure), so basically, totally off-base. Sorry.

Simon: Hmmm. You might be right. The simplest macro (e.g. Solow growth) technology has two production functions: Y=F(K,L) (2 inputs one output); and C+Kdot=Y (1 input 2 outputs that are perfect substitutes in supply, so they are really only one good).

Financial engineering is another area where the many-output model is the standard. The simplest case is simply taking a standard coupon bond and selling rights to the coupons (an annuity) and the face value (a pure discount bond.) CDOs went much further, pooling assets (multiple inputs) and selling tranches of the pool that differed by seniority (multiple outputs).

two hats: no worries. Actually, it's because economists have axes sort of switched, because when you draw a demand or supply curve, P is the independent variable (which should normally be on the horizontal), and Q the dependent (which should normally be on the vertical). Historical accident, and it's too late for us to switch now!

Joel W: Aha! So Varian still has a bias towards one output many inputs!

"No disrespect, but those are "weird" cases where there is non-excludability."

Think of, e.g., a music distributor posting tunes on his/her website, and working out the potential revenue to be gained from each unique tune. Each individual consumer has a demand for tunes of Pi=a-bX where X=number of unique tunes downloaded. The math is the same as the case you discuss, whether or not the music distributor can charge for downloads.

In other words, non-rivalry is what generates the vertical summation - non-excludability is what causes the particular type of market failure seen with externalities and public goods.

"It's people's preferences for meat and wool, and not just the technology for producing meat and wool, that are needed to determine the prices of meat and wool."

Also distortions in one market spill over into another, i.e. a meat cartel impacts the wool market, and vice versa.

Helium is another really fascinating example - produced as a byproduct of, I think, natural gas?

I'm am studying "Behaviour Change Theory", partly to figure out why people won't stock a pantry for pandemics. It delves into the human capital metric. Some things people just like doing and for some things, the cash incentive is nice. And I'm not too happy with the $16/hr value estimate of volunteer time social capital estimates came up with. B.Gates aside, the value of people's volunteer time is probably much less than their earned wage. The Jobs Act that was 7 USA Senate votes short of passing last fall, was funded in part out of charities; though they have the NRA as biggest one...
I'm curious if behaviour and cash + time are mostly substitutable. If so, there is big money in surveys/stats and survey/stats companies/Crowns. Offering welfare or a GAI in exchange for health nutrition info or some minor FDA tests is probably cost effective...considering children: yes. Considering adults: maybe only sometimes can you pay people to be prosperous.

Sheep *are* neat!!

Nick, you may be looking at this from a different direction (probably?) but there is a modetn IO literature on economies of scope and multi-product firms going back to the late '70s early 80' - Panzar and Willig.

Nick: As SvN says, this is mostly an undergraduate question--in graduate micro we tend to think in terms of netput vectors. At the undergraduate level, the answer might be mostly one of path dependency in the history of thought, but I think there is a deeper reason. The multiple input single output world leads naturally into solving with a dynamic programme (not that we typically call it that). That is, instead of solving the simulatenous problem of choosing inputs and output to maximise profit, we break it down into two steps: First, minimise cost for a given output to find a cost function, and using backward induction choose output to maximise revenue minus the cost given by that function. That cost function is awfully useful for deriving firm supply and producer surplus, illustrating monopoly, etc. I don't think that single-input, multiple-output examples provide such a pedagogically useful dynamic programme.

That said, in my intermediate micro class, I teach multiple-input/single-output and then give a problem set with the reverse.

Stop me if you've heard this one before...

How many units of labour does it take to screw in a lightbulb?

/rim shot/

Off the main topic, but relevant to the post, is the issue of how we individuate and count inputs (or outputs). The same problem comes up in ecology, although most ecologists don't recognize it. A very general theoretical principle in ecology is that, if you have a bunch of species competing for resources (in economics terms, competing for the various "inputs" to their "production functions"), you can't have more species stably coexisting at equilibrium than there are distinct resources. Which ecologists traditionally regard as a puzzle, since nature seems to confront us with many exceptions to this principle. For instance, tropical rainforests have many hundreds of coexisting tree species, even though all trees need the same relatively short list of resources (water, light, nitrogen, phosphorus). But as Peter Abrams (recently retired from UToronto) pointed out in a sadly-neglected paper, thinking of this as a puzzle assumes that we know how to count resources from the perspective of the trees. For instance, maybe trees differ in their rooting depths, so that nitrogen from deep underground is a different resource than nitrogen from near the soil surface. Abrams discussed what would need to be the case for two things to count as different resources. All this leads me to wonder sometimes about the conditions that cause inputs to be more or less finely subdivided...

I read this and immediately thought of oil refining, which produces hundreds of products


Why aren't production functions with multiple outputs as important to us as production functions with multiple inputs?

Because we can usually reduce this to the case of a firm selling only a single output. We can imagine a store that sells beer and wine to be two different stores that sell beer each or wine each.

Similarly, we can conceive of the farm selling lamb wool and lamb meat as two different businesses, one that rents the sheep and shears it. The depreciation of the sheep corresponds to the loss of value or other harm to the sheep as a result of being sheared. Then another firm buys the sheep and extracts meat. In principle, there is no reason why all sheep need to have their wool sold. One can imagine that some sheep are only used for meat, while others are only used for wool.

But on the other hand, the producer really does use inputs that must be composite. Every producer, for example, will need to hire an accountant, rent land, buy electricity, purchase tools as well as intermediate inputs that are all necessary to the production process.

Jeremy: doesn't that assume that two species of tree only interact through competing for the same inputs? (Suppose both trees (OK plants) need bees, and the bees need both types of tree, because they flower at different times?)

Marris: great example.

rsj: I can only divide a beer and wine store into two separate stores of the production function looks like this: Beer sold + wine sold = F(inputs), so the two outputs are perfect substitutes. If there are complementarities of selling both, you cannot split them into two production functions. It's non-separable.

Linda: good point. I had forgotten the IO literature on economies of scope. Sheep are just an extreme example of economies of scope.

Seamus: good argument. But I wonder: oughtn't it work symmetrically, somehow? After all, what's the difference, mathematically, between a function with 1 Y and 2 X's and a function with 1 X and 2 Y's? The math doesn't even know that Y is an output and X is an input?

Frances: "In other words, non-rivalry is what generates the vertical summation - non-excludability is what causes the particular type of market failure seen with externalities and public goods."

Ummm. That is a nice intuition. Meat and wool are non-rival.

rsj: I can only divide a beer and wine store into two separate stores of the production function looks like this: Beer sold + wine sold = F(inputs), so the two outputs are perfect substitutes. If there are complementarities of selling both, you cannot split them into two production functions. It's non-separable.

So we cannot talk about a stores that sells gas and another store that sells cars? Every gas station must also sell cars, and every car dealership must sell gas?

Or perhaps I don't understand what is meant by "complementarities of selling". It seems to me that businesses can sprout up and sell whatever they want. Nothing prevents a conglomerate from selling ramen noodles and cranes -- look at Japan. At the same time, nothing prevents a company from selling, say, art pencils and not school pencils or art paper. It might be that the firm that dominates the market in art pencils is different from the firm that dominates the market in regular school pencils, even though the production process for making school pencils is very similar to the production process for making art pencils.

I don't see any connection between production processes and firms per se.

Hi Nick:

Yes, I'm assuming that the trees interact only via competing for those shared resources (as opposed, to, say, one tree producing toxins that kill other trees).

In your example of plants that would compete for bee pollinators, except that they flower at different times, the same bees at times t1 and t2 constitute two different resources, with each plant specializing on one of them.

In general, it's surprising how organisms can manage to subdivide seemingly indivisible resource inputs. There are marine algae that specialize on different wavelengths of the visible light spectrum, so that "light" is not one single resource for them--instead they differ in their relative abilities to use red light vs. green light vs. yellow light.

This is one of those areas where there are lots of interesting overlaps between ecology and economics, but also interesting contrasts. For instance, ecologists consider all of the same sorts of production functions economists do (because all those production functions actually show up in real organisms), but we have different names for them. And we tend to use those production functions to ask different questions. For instance, it turns out that the conditions for stable coexistence of multiple competing species differ somewhat, depending on what production function each species uses. David Tilman once argued that this is the ultimate explanation for why there are so many species of animals (many millions) and so few species of plants (well under a million)--different production functions for animals vs. plants. But it's a difficult claim to test, in large part because (to bring it back to my original comment) we usually can't be sure that we're individuating resources the way the organisms do. We tend to think of animals as competing for perfect or near-perfect substitutes (e.g., wildebeasts and zebras are basically substitutable for lions), and plants competing for what ecologists call essential or non-substitutable resources (e.g., water can't substitute for light, and vice versa, and neither can substitute for nitrogen). But it's not clear that that's actually true (e.g., those marine algae that see different wavelengths of light as substitutable resources). And in some of my own work, I've shown how natural selection acting on competing species has different consequences, depending on the production functions of those competitors.

Ah but I think you are forgetting the most common cases of your multiple output production process... and it applies to your scotch example too: price discrimination: e.g., scotch bought at the airport is distinct from scotch bought at the distillery if it it was bottled from the same cask at the same time.

IANAE, so feel free to ignore, but I've always wondered how prices for the different foods in restaurants get chosen, or how prices are allocated to food and wine. Is this the same question?

If I recall correctly, Donald McCloskey's textbook The Applied Theory of Price discusses joint outputs at some length and has a diagram like one you posted.

Jeremy, interesting comments.

Multiple outputs for one input have a long history in the regulated public utility industry for purposes of cost recovery. They include common cost for variable outputs as well as joint cost for fixed outputs. The essential problem is that cost causality cannot be readily identified for individual services and rates, and therefore must be calculated on a shared cost basis using something like "Shapley Values" or other formulas.

For example landline telephone service was an ongoing conflict for how rates should be set to recover common and/or joint cost depending on chosen definitions. For multiple output services like local, long distance, call waiting, caller id, etc, phone companies would insist for ratemaking purposes that there was only one input service called "access" which effectively eliminated the possiblity of most underlying joint or common cost for other services.

The purpose of course was to exploit the monopoly power ability to load and shift most cost recovery into regulated basic "access service" then report sharply reduced underearnings for tangential services like caller id which were being deregulated as "non-essential" services.

In non-regulated areas where competition truly exists that prevents economic profit and allows only normal profit and loss, the notion of common and joint cost recovery comes down to price elasticities of the full batch of true end use services that arise from each (despite the role of elasticity in monopoly power in general).

For example competitive supermarkets would allocate fixed input costs like freezers and parking lots across thousands of products on the shelves while non-competitive supermarkets would single out these costs and charge for them separately as individual services on a stand alone basis the way telephone companies do for landline "access service".

Nick: I can only divide a beer and wine store into two separate stores of the production function looks like this: Beer sold + wine sold = F(inputs), so the two outputs are perfect substitutes. If there are complementarities of selling both, you cannot split them into two production functions. It's non-separable.

If the beer store and wine store are owned by different people, we call that an externality. Obviously it's internalized if the beer store owner and wine store owner are the same person, but why call it something different?

"But why do only environmental economists and public finance economists talk about the case of joint products?"

Because they have no recourse to the market, which rolls all of these possibilities into a single price.

My interpretation: business is by nature synthetic. Goods and labor are converted into a single (intended) outcome, profit. How people cook their pork makes no difference to the farmer. Or perhaps it does - he might innovate a product that better captures their demand - but it makes no difference to the economist, who cannot know what the farmer thinks or what he might innovate or whether his innovation will succeed, but only what he has actually sold.

All an impartial observer has is the price at which the transaction(s) occurred. We can talk about subdivided demand curves, or the existence of sub-particles beyond our comprehension, but the fact that a thing is possible, even probable - or for that matter, publishable - doesn't make it scientific. It must be measurable.

Nor can we prove anything by watching where outputs end up. Demand is a function of intended, not actual use. Some fraction of tomatoes always gets tossed, but this doesn't mean there is demand (or a demand curve) for rotten tomatoes. Common sense tells us that these are generally bought to be eaten; statistics tells us how many are actually eaten; exit polling tells us what some people said after the fact; but we don't know anything.

What you're seeing is, I think, the result of a (very justified) distinction between knowledge and inference.

Perhaps regulatory bodies and central planners view the process differently. Working backward from outputs they regard as more or less desirable, they attempt to divine the proper input. But this is a moral or philosophical exercise, not an economic one, whatever terminology (and however many economists) they might employ. I suppose a taxidermist needs some knowledge of biology in order to ply his trade -- but this doesn't make him a biologist.

My Applied Linear Algebra text has the line: "The same form of equations will hold if we have p different machines producing q products." (where different combinations of machines are required for each product.) "Then A is a p x q matrix, ..."

As far as I can tell, this makes hash out of simple theories of specialization and comparative advantage, since specialization would seem to be generally an inefficient use of multiple inputs, and optimum is some suite of products. Consider a modern economy: p different machines producing q products.

Of course, one can imagine the case of machines so specialized they are each only good for just one product, but is this the usual case?

@Nick: "But I wonder: oughtn't it work symmetrically, somehow? After all, what's the difference, mathematically, between a function with 1 Y and 2 X's and a function with 1 X and 2 Y's? The math doesn't even know that Y is an output and X is an input?"

Yes, everything you can do with 1 output and 2 inputs has an analogue with 2 outputs and 1 input. The question is whether those analogues are interesting. So you can do the dynamic programme by finding revenue as a function of the input, and showing that Marginal revenue below Average revenue is the input demand function; you get N-shaped average revenue if you assume that the limit of revenue as input tends to zero is negative (WTF?); free entry gets you an approximately horizontal input demand function at the maximum of this N-shaped average revenue curve (maybe not so bad); you can do some Dixit-Stiglitz type analysis if you assume that every firm looks a bit different to workers so that there is monopsonistic competition and possibly too much labour market diversity in equilibrium (hmm), and so on. The question is whether these are as interesting as their analogues. Now perhaps this is just begging the question, as you would want to know why these applications are less interesting, but it perhaps suggests where to look for the answer.

One answer would be that the choice between two inputs in particular--labour and capital--is at the heart of questions about economic welfare in the short and long run. This is not likely to be true about a distinction between two outputs.

Good comments all. I have been reading, honest! But just can't think of anything worth saying in response.

In one of the managerial econ books I've used (it may be Mike Baye's book, but I don't remember for sure, and I don't have any of the books at hand), there's a beautiful table illustrating all of the various parts of cattle that are harvested and sold for various purposes. This in the context of joint production, one of the points of which is the amount of (say) collagen supplied from cattle (which in the analysis in whichever book this is in) is treated as being supplied at zero cost (the supply curve of gollagen is vertical) depends on the demand for the *primary* product being produced.

And if collagen becomes the primary product, then the supply curve of *beef* becomes vertical...

Minor observation: *Bourbon* whiskey ageing in a cask is like sheep: inputs are raw whiskey and a new barrel, outputs are aged bourbon and a used barrel that may be purchased for use in ageing Scotch. This is a relatively new development, the Scotch makers mostly used sherry barrels until declining sherry consumption led to a barrel scarcity. (You could also argue that there is an input of a scenic location and an "output" of distillery tourism)

I suspect, in general, production processes tend to drift towards multiple outputs simply as a function of trying to increase efficiency. Besides the obvious tack of trying to use less input per unit of output you try to find ways of turning what would otherwise be waste/uused into an alternate product.

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